Quote:
In probability theory, the Kelly criterion, or Kelly strategy or Kelly formula, or Kelly bet, is a formula used to determine the optimal size of a series of bets. Under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run, with probability 1.
http://en.wikipedia.org/wiki/Kelly_criterion
In a universe of +EV bets, the Kelly Criterion is a mathematical formula used to size bets for optimal bankroll growth. Given that you know the current size of your bankroll, the odds on your bet, and an estimate on the probability on your bet winning, it tells you exactly what % of your bankroll you need to bet in order to grow your bankroll the fastest. It is used widely in the investing world, and sharp gamblers use it for sports betting as well.
One of the reasons that we don’t hear about the tool in poker is that usually we are more preoccupied with other concerns – most notably our opponent’s reaction to our decisions, but also the playability of hands, metagame, planning for later streets, etc. Especially damning is the standardized size of bets in Limit Holdem. However, there are a few situations where you do have some control over your bet-sizing, the most notoriously debated on this forum is that of whether or not to raise with a “marginal” appearing holding when it is folded to you in one of the two blind spots. You can choose to check and wager nothing or raise and wager 1 SB.
Example: You are playing in a normal 20/40 game and have QT
in the BB.
There are 3 limpers to you and the SB completes. Your move.
The assumed ranges are appended below in the Poker Stove simulation, but the summary is that the limper #1 is assumed to have the top middle 30.3% if his range, limper #2 -37.3%, limper #3 - 44.5%, and the SB – 87.6%. You can argue with these assumptions quite a bit without changing the primary result – our equity is estimated at 25.8%
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25,456,273 games 53.375 secs 476,932 games/sec
Board:
Dead:
equity win tie pots won pots tied
Hand 0: 25.850% 24.12% 01.73% 6139937 440547.77 { QTs }
Hand 1: 19.894% 18.41% 01.48% 4686334 377824.93 { AhAs, KhKs, 88-55, A9s-A2s, K6s+, Q8s+, J8s+, T7s+, 96s+, 86s+, 75s+, 65s, 54s, A9o-A7o, K8o+, Q9o+, J8o+, T8o+, 97o+, 87o, 76o }
Hand 2: 18.732% 17.35% 01.39% 4415755 352710.10 { 88-22, A9s-A2s, K5s+, Q7s+, J7s+, T6s+, 96s+, 85s+, 75s+, 64s+, 54s, 43s, A9o-A6o, KJo-K8o, Q8o+, J8o+, T7o+, 97o+, 86o+, 76o, 65o }
Hand 3: 18.709% 17.39% 01.32% 4426374 336145.02 { 88-22, A9s-A2s, K2s+, Q5s+, J6s+, T6s+, 95s+, 85s+, 74s+, 64s+, 53s+, 43s, A9o-A4o, KJo-K7o, Q8o+, J8o+, T7o+, 97o+, 86o+, 75o+, 65o, 54o }
Hand 4: 16.816% 15.73% 01.09% 4004091 276554.18 { TT-22, AJs-A2s, K2s+, Q2s+, J2s+, T2s+, 92s+, 82s+, 72s+, 62s+, 52s+, 42s+, 32s, AQo-A2o, K2o+, Q2o+, J2o+, T2o+, 95o+, 84o+, 73o+, 63o+, 53o+, 43o }
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Plugging these variables into a
Kelly Calculator(5-1 odds, 25.8% chance of winning) recommends that you should wager 10.96% of your bankroll if allotted.
Of consideration is that the Kelly Criterion concerns itself primarily with maximal bankroll growth. Reducing risk of ruin is an implicit consideration but a secondary goal. Thus, it is a common procedure to reduce the amount Kelly advises, the most common recommendation being one-half, also known as half-kelly betting. In this situation Half-kelly would still call for 5.48% of your bankroll. For a 300 BB(600 SB) bankroll that would be equivalent to ~32 SBs. (This figure may seem big to you, but it takes into account the factor below, and also is a comment on how antediluvian our standard bankroll requirements are.) To not put in 1 more SB would be a ridiculous undersizing of a bet and to want to reduce variance is at the expense of a significant amount of expected growth(EG).
The Kelly Criterion also assumes that all of your bets are not correlated with any other bets(covariance = 0). While it is true that your decision to raise or check has a small to no change on whether you are more likely to win the hand, you will be making more bets on the same hand for 3 more streets. Therefore, it would be prudent to make sure the maximum amount you are wagering during the hand is still within the confines of the recommended size. However, there are few to no hands you play with QTs that you should expect to lose 16BBs.
At this point, it’s worth examining the other factors that you will be subjected to if you decide to raise – you will be playing the hand out of position and are increasing the size of the pot with initiative. This obviously makes hands with good implied odds and can allow your opponents to draw thin or incorrectly(big paint suited connectors) much better than to raise with than hands that are the opposite of that(unsuited big aces). There are other considerations at play, but you can find them in a number of other old threads in this forum debating preflop. There are of secondary importance here. (I will add that most people argue irrationally against raising in these spots because they are uncomfortable with checking certain flops, check-raising certain flops, etc.)
If you are anywhere near the recommended bankroll for the game you are playing in, it is an absolutely gross underbet to check a lot of these hands we routinely debate in this forum from a growth perspective. It’s not even close. You may feel relieved when the board comes blank and you get to “save a bet”, and just as happy when you turn your flushes, but your bankroll will suffer immensely in the long run. Though your play will still be +EV, it will often be –EG.
The Bryce was right when he proclaimed that “guessing is out and math is king”. Once you’ve attained a level of competence in LHE, it’s important to examine the rule of thumbs you use every time you play and wonder if they are compounding errors that are compromising the growth of your bankroll. What you feel is something you want to apply to reads, not when you are doing a math problem. Being concerned only with +EV is so last season anyways. The future is all about +EG.